The Number of Real Roots of a Cubic Equation

The Number of Real Roots of a Cubic Equation Richard Kavinoky Santa Rosa Junior College J. B. Thoo Yuba College November 6, 2007 The number of distinct real roots 1 of the cubic equation x + bx 2 + cx + d = 0 (1) equals, for example, the number of normals to the parabola y = x 2 through a given point in the plane (Bains and Thoo (2007)), 2 as well as the number of equilibrium solutions of dx/dt = x + bx 2 + cx + d. Now, one way to find the number of real roots of (1) is to solve the equation. Certainly, (1) can be solved by hand easily if d = 0. If d 0, the equation can still be solved by hand using Cardan s (or Cardano s) cubic formula (Fine (1961); Gellert et al. (1975)), but not easily in general. Of course, using computer software or a graphing calculator can make light work of solving (1) altogether. However, there is a certain satisfaction in being able to tell the number of real roots of (1) without first solving the equation. It turns out that this is easy to do. The key is the discriminant. Every intermediate algebra student learns the quadratic formula, deriving it by completing the square. And with the quadratic formula in hand, it is apparent that the number of real roots of the quadratic equation x 2 +bx+c = 0 1501 Mendocino Avenue, Santa Rosa, CA 95401-42, rkavinoky@santarosa.edu. 2088 N. Beale Road, Marysville, CA 95901-7605, jthoo@yccd.edu. 1 We mean the number of distinct real roots throughout. 2 It is remarkable that Apollonius had obtained precise results on the number of normals to a parabola through a given point using purely synthetic geometry (Heath, 1981, pp. 158 159, 16 166) almost 1700 years before Italian mathematicians solved the cubic. For Cardan s formulation of the solution of the cubic, as well as the story behind the quarrel between Cardan and Tartaglia over its publication, see Burton (200) for example. For other formulations of the solution of the cubic, see Kalman and White (1998) and the references therein. 1

is determined by its discriminant, b 2 4c. But very few students today see Cardan s cubic formula, and its derivation is much less straightforward than that of the quadratic formula. So, how may a student today come up with or be led to the discriminant of the cubic equation (1) without appealing to the cubic formula? In this note, we present one way of doing so using ideas from a first calculus course derivative, critical point, local extrema, and graphing in an intuitive way. We also show how the discriminant defines a boundary in the plane across which the number of real roots of (1) changes, and apply the discriminant to determining the number of normals to the parabola y = x 2 through a given point and the number of equilibrium solutions of dx/dt = (R R c )x ax, the Landau equation in fluid mechanics (Boyce and DiPrima, 200, p. 89), where R c and a are positive constants and R is a parameter. Discriminant of a cubic equation A good way to begin is by looking at the graph of y = x + bx 2 + cx + d. To simplify the analysis, we translate the graph horizontally through the change of variables x x b/, thereby obtaining the graph of y = x + px + q that has its inflection point on the y axis, and examine equivalently the number of real roots of the reduced cubic equation x + px + q = 0. (2) Toward this end, we define the function C(x) = x + px + q and consider the following cases. I: p 0 x + px + q = 0 II: p < 0 A: q = 0 B: q < 0 C: q > 0 Case I: p 0 2

If p = 0, then C(x) = x + q, so (2) has one real root, namely, q 1/. Otherwise, C (x) 0 for all x, so C is monotone increasing and, hence, (2) again has one real root. Case II.A: p < 0 and q = 0 Then C(x) = x + px, so (2) has three distinct real roots: ( p) 1/2, 0, and ( p) 1/2. For the next two cases, we note that the critical points of C are ) 1/2 x = and x + = ) 1/2, and that the second derivative test implies that C(x ) is a local maximum and C(x + ) is a local minimum. Case II.B: p < 0 and q < 0 Then C(0) < 0. This gives three possibilities for the graph of y = C(x); see Figure 1. It follows that (2) has one real root if and only if C(x ) < 0; two distinct real roots if and only if C(x ) = 0; and three distinct real roots if and only if C(x ) > 0. x x + x x + x x + (a) C(x ) < 0 (b) C(x ) = 0 (c) C(x ) > 0 Figure 1: Case II.B: p < 0 and q < 0. First suppose that C(x ) < 0. Using p we find that C(x ) = ) 1/2 = ( p) 2/2 ) /2 p ) 1/2 ) /2, = ) 1/2 ) /2 + q = 2 + q.

Thus, C(x ) < 0 if and only if q ) /2. 2 < Since both sides of this are negative, squaring yields the equivalent inequality q 2 ), 4 > which is equivalent to 4 + p 27 > 0. Therefore, (2) has one real root if and only if q 2 /4 + p /27 > 0. q 2 Next suppose that C(x ) = 0 or C(x ) > 0. A similar analysis shows that (2) has two distinct real roots if an only if q 2 /4 + p /27 = 0, and three distinct real roots if and only if q 2 /4 + p /27 < 0. Case II.C: p < 0 and q > 0 Then C(0) > 0. This again gives three possibilities for the graph of y = C(x); see Figure 2. Now it follows that (2) has one real root if and only if C(x + ) > 0; two distinct real roots if and only if C(x + ) = 0; and three distinct real roots if and only if C(x + ) < 0. x x + x x + x x + (a) C(x + ) > 0 (b) C(x + ) = 0 (c) C(x + ) < 0 Figure 2: Case II.C: p < 0 and q > 0. Following the analysis of Case II.B, we find that (2) has one real root if and only if q 2 /4 + p /27 > 0; two distinct real roots if and only if q 2 /4 + p /27 = 0; and three distinct real roots if and only if q 2 /4 + p /27 < 0. What we have shown, therefore, is that the number of real roots of the cubic equation (1) is characterized by D = q2 4 + p 27, 4

called the discriminant of the equation, where p = c b2 and q = 2b 27 bc + d. We summarize this in the following proposition. Proposition 1 If p = q = 0, then the cubic equation x + bx 2 + cx + d = 0 has one root, namely, zero. Otherwise, the cubic equation has one real root if and only if D > 0. two distinct real roots if and only if D = 0. three distinct real roots if and only if D < 0. We remark that the discriminant D = q 2 /4 + p /27 appears in Cardan s cubic formula. Geometry of the discriminant The discriminant q 2 /4 + p /27 gives some insight into the number of real roots of (1). Specifically, the graph of q 2 /4+p /27 = 0 is a boundary in the qp plane across which the number of real roots of (1) changes in the same way that the graph of b 2 4c = 0 is a boundary in the bc plane across which the number of real roots of a quadratic equation x 2 + bx + c = 0 changes. See Figure. Two applications We apply the discriminant D = q 2 /4 + p /27 to determining the number of normals to the parabola y = x 2 through a given point and the number of equilibrium solutions of the Landau equation dx/dt = (R R c )x ax. A normal to the parabola y = x 2 through the point (α, β) satisfies the relation (y β)/(x α) = 1/(2x) that simplifies to the reduced cubic equation (2) with p = 1 2β 2 and q = α 2. Thus, given (α, β), Proposition 1 tells us precisely the number of normals to the parabola y = x 2 through the point. Better yet, if we set D = 0 and use the 5

10 p 5 D > 0: one real root 10 c 5 D < 0: no real root 0 0 5 D < 0: three real roots 5 D > 0: two real roots 10 10 5 0 5 q 10 (a) Cubic equation: D = q 2 /4 + p /27 10 10 5 0 5 b 10 (b) Quadratic equation: D = b 2 4c Figure : The graph of the discriminant D = 0 is a boundary across which the number of real roots changes. above relations for p and q with α = x and β = y, we obtain the function y = 2 4/ x2/ + 1 2 = N(x). Figure 4 shows the graph of y = N(x), a semicubical parabola just like the boundary curve in Figure (a), together with the parabola y = x 2. The semicubical parabola is often called Neil s (or Neile s) parabola (Bains and Thoo (2007); Gellert et al. (1975)). Geometrically, then, there is one normal to the parabola y = x 2 through the point (α, β) if the point lies below or on the cusp of Neil s parabola y = N(x); two normals if the point lies on Neil s parabola, except on the cusp; and three normals if the point lies above Neil s parabola. Turning to the Landau equation dx dt = (R R c)x ax, () where R c and a are positive constants and R is a parameter, an equilibrium solution is one such that dx/dt = 0. Thus, an equilibrium solution of () is a solution of the reduced cubic equation (2) with p = R c R a and q = 0. Proposition 1 then implies that () has one equilibrium solution if R R c and three equilibrium solutions if R > R c. Further, the proposition implies that () cannot have two equilibrium solutions. Geometrically, the point (0, (R c R)/a) 6

4 2 1 0 1 2 1 0 1 2 Figure 4: Neil s parabola (heavy curve) together with the parabola y = x 2. lies above or on the cusp of the boundary curve shown in Figure (a) if R R c, and (0, (R c R)/a) lies below the curve if R > R c. The value of the parameter R = R c is called a bifurcation point. Note that here the bifurcation point corresponds to (0, 0), the cusp of the boundary curve. More generally, if we modify the Landau equation () by adding a nonzero constant b to the right-hand side, dx dt = b + (R R c)x ax, then Proposition 1 implies that the equation has one equilibrium solution if R < R c + ab 2 /4 and three equilibrium solutions if R > R c + ab 2 /4. But, unlike (), the modified equation can also have two equilibrium solutions. This occurs if R = R c + ab 2 /4. Note that, because b 0, the point ( b/a, (R c R)/a) does not pass through the cusp of the boundary curve in Figure (a) as the parameter R varies. References Bains, M. S. and Thoo, J. B. (2007). The normals to a parabola and the real roots of a cubic. Coll. Math. J., 8(4):272 277. Boyce, W. E. and DiPrima, R. C. (200). Elementary Differential Equations and Boundary Value Problems. John Wiley & Sons, Inc., New Jersey, 7th edition. 7

Burton, D. M. (200). The History of Mathematics: An Introduction. McGraw- Hill, Boston, 5th edition. Fine, H. B. (1961). College Algebra. Dover Publications, Inc., New York. Gellert, W., Kustner, H., Hellwich, M., and Kastner, H., editors (1975). The VNR Concise Encyclopedia of Mathematics. Van Norstrand Reinhold Co., New York. Heath, T. (1981). A History of Greek Mathematics Volume II: From Aristarchus to Diophantus. Dover Publications, Inc., New York. Kalman, D. and White, J. (1998). A simple solution of the cubic. Coll. Math. J., 29(5):415 418. 8